Littlefinger theory goes like this....
LittleFinger early in the series spoke of having family (Grandfather) from braavos - the home of the Faceless Men. Remember that him being the Master of Coin meant LF regularly frequented the Iron Bank in Braavos so he would be very, very well aquinted with the area and its people.
LF "likes to play a game...to see the worst outcome.. blah blah". Bran the 3 Eyed Raven, reiterates back to LF his own quote, "Chaos is a ladder", which he'd told Ned at the time of betraying their father. Many assume that it would've been at this point that LF would have a 'heads up' that something is/was very very wrong and in true LF style, begin to make his preperations.
There one scene further down the line where Ayra is spying on LF and she see's him talking in secret to a young woman in a barn (or something similar). She whispers, "your time's up" and he then gives her a silver looking coin. Who was she and why were they talking..? It's theorised that because she didn't look like anyone of importance (she is dressed like a kitchen worker or general helping hand), nor do we (the viewers) recognise her, it's possible she is a Faceless Assassin. Furthermore nothing else is referenced in later episodes of their meeting, so it's kind of left open...
The silver looking coin could be an iron coin - like the one Ayra gave to the captain to allow boarding of his ship to Braavos, and it could very well have been payment to a Faceless Man to take his place if LF's suspicion of Stark revenge for their Father turns out to be true.
LittleFinger is too sneaky to have been caught out as he (apparently) did and whilst everyone is squabbling amongst themselves for the IT and for survival against the WW and the NK, no one would ever see him coming. Everyone kills themselves off in the struggles and he emerges from the shadows to take the throne - considering his prior role as Master of Coin and as member of the High Council - he would be the most 'qualified' to rule after most houses have been depleted of their heirs.
I've given this much thought! 